Sparse signal recovery deals with finding the sparest solution of an under-determined linear system $x = Qs$. In this paper, we propose a novel greedy approach to addressing the challenges from such a problem. Such an approach is based on a characterization of solutions to the system, which allows us to work on the sparse recovery in the $s$-space directly with a given measure. With $l_2$-based measure, two OMP-type algorithms are proposed, which significantly outperform the classical OMP algorithm in terms of recovery accuracy while maintaining comparable computational complexity. An $l_1$-based algorithm, denoted as $\text{Alg}_{GBP}$ (greedy basis pursuit) algorithm, is derived. Such an algorithm significantly outperforms the classical BP algorithm. A CoSaMP-type algorithm is also proposed to further enhance the performance of the two proposed OMP-type algorithms. The superior performance of our proposed algorithms is demonstrated through extensive numerical simulations using synthetic data as well as video signals, highlighting their potential for various applications in compressed sensing and signal processing.

翻译：稀疏信号恢复旨在从欠定线性系统 $x = Qs$ 中寻找最稀疏解。本文提出了一种新的贪心方法以应对此类问题带来的挑战。该方法基于系统解的一种表征形式，允许我们利用给定度量直接在 $s$ 空间上进行稀疏恢复。基于 $l_2$ 度量，我们提出了两种 OMP 型算法，这些算法在恢复精度上显著优于传统 OMP 算法，同时保持相当的计算复杂度。我们还推导了一种基于 $l_1$ 的算法，记为 $\text{Alg}_{GBP}$（贪心基追踪）算法。该算法显著优于传统 BP 算法。此外，还提出了一种 CoSaMP 型算法，以进一步提升所提两种 OMP 型算法的性能。通过使用合成数据及视频信号的大量数值模拟，我们展示了所提算法的优越性能，凸显了它们在压缩感知和信号处理中的潜在应用价值。